This question was previously posted to Computer Science Stack Exchange here.

Imagine you're a very successful travelling salesman with clients all over the country. To speed up shipping, you've developed a fleet of disposable delivery drones, each with an effective range of 50 kilometers. With this innovation, instead of travelling to each city to deliver your goods, you only need to fly your helicopter within 50km and let the drones finish the job.

More precisely, given a real number $R>0$ and $N$ distinct points $\{p_1, p_2, \ldots, p_N\}$ in the Euclidean plane, which path intersecting a closed disk of radius $R$ about each point minimizes total arc length? The path need not be closed and may intersect the disks in any order.

Clearly this problem reduces to TSP as $R \to 0$, so I don't expect to find an efficient exact algorithm. I would be satisfied to know what this problem is called in the literature and if efficient approximation algorithms are known.

• Also posted on CS.SE. Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. If you don't get a satisfying answer after a week or so, feel free to flag for migration. – D.W. May 17 '15 at 22:06